We characterize which sets of k points chosen from n points
spaced evenly around a circle have the property that, for each
i = 1, 2, …, k − 1,
there is a nonzero distance along the circle that occurs as the distance
between exactly i pairs from the set of k points.
Such a set can be interpreted as the set of onsets in a rhythm of
period n,
or as the set of pitches in a scale of n tones, in which case the
property states that, for each
i = 1, 2, …, k − 1,
there is a nonzero time [tone]
interval that appears as the temporal [pitch] distance
between exactly i pairs of onsets [pitches].
Rhythms with this property are called Erdős-deep.
The problem is a discrete, one-dimensional (circular) analog to
an unsolved problem posed by Erdős in the plane.